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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
0
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1
answer
161
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Sufficient and necessary condition for the global uniqueness of fix-points
https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf
This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if the …
1
vote
0
answers
385
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The perturbation of a convex function can also be convex?
$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a convex …
5
votes
1
answer
305
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Reference for the rectifiablity of the boundary hypersurface of convex open set
The boundary of any convex open set $X$ is $\mathbb R^n$ is a rectifiable hypersurface.
To see this, intuitively, simply take a sphere $S_d$ with diameter $d\in(0,+\infty]$ that contains $X$. The …